PROJECT REPORT

DENSITY FUNCTIONAL THEORY

BY

AFNAN BAKHT BAHROZ RASHID WAQAS LIAQAT NOMAN KHAN

SHEHRISH AKBAR

INSTITUTE OF PHYSICS & ELECTRONICS UNIVERSITY OF PESHAWAR

1

MANY BODY PROBLEM

Many body problem is a general name given to the properties of quantum system made of

large number of interacting particle. The wave function of such a system is complicated and

holds a large amount of information. In such a quantum system, the repeated interactions

between particles create quantum co -relation or entanglement. The main problem of such

system is to find the ground state wave function of interacting electrons with an external local

potential. The solution of such system relies on approximations specific to system at hand.

EXAMPLES:

Condensed matter physics (solid-state physics, nanoscience, superconductivity)

Quantum chemistry (computational chemistry, molecular physics)

Atomic physics

Molecular physics

Nuclear physics (Nuclear structure, nuclear reactions, nuclear matter.

APPROACHES:

Out of many approaches to solve the many body problem, we will use Density

Functional Theory

2

DENSITY FUNCTIONAL THEORY

Density functional theory is an approach for the description of ground state properties of

metals, semiconductors, and insulators. The success of density functional theory (DFT) not only

encompasses standard bulk materials but also complex materials such as proteins and carbon

nanotubes. The main idea of DFT is to describe an interacting system of fermions via its density

and not via its many-body wave function. For N electrons in a solid, which obey the Pauli

principle and repulse each other via the Coulomb potential, this means that the basic variable

of the system depends only on three -- the spatial coordinates x, y, and z rather than 3N

degrees of freedom.

SIGNIFICANCE:

DFT is among popular and versatile methods available in condense matter physics,

computational physics and computational chemistry. The wave function of an N-electron

system includes 3N variables, while the density, no matter how large the system is, has only

three variables x, y, and z. Moving from E*+ to E*+ in computational chemistry significantly

reduces the computational effort needed to understand electronic properties of atoms,

molecules, and solids. The results of DFT calculation for solid state system are according with

the experimental data.

3

DERIVATION AND FORMALISM

To find ground state of N Interacting electrons in an external potential of nuclei, we solve the

Schrdingers equation

Where is Hamiltonian operator corresponding to the total energy of the system. In this case,

it is equal to the sum of total kinetic energy operator of the electrons, Coulomb interaction

between the electrons and external potential which is between the electrons and

nuclei. Now we apply a well know Born Oppenheimer approximation [01] on this Hamiltonian,

in which nuclei are taken at rest as compared to fast moving electrons, so the external potential

become static. The Hamiltonian becomes:

(

)

( )

Now from first equation

((

)

( )

)

This is a second order differential equation with 3N independent co-ordinates and is extremely

difficult to handle. We use alternative methods to solve this problem.

4

RITZ VARIATIONAL PRINCIPAL

For solving the second order differential equation with 3N independent co-ordinates we use

Ritz variational principal [02] to find the ground state of the system. In this approach, we

minimize the ground state energy of N particles.

| |

This is done by first setting up the Hamiltonian operator of target system. From equation (1.2) it

shows that the only information that depends on the actual molecule is the number of

electrons in the system N and the external potential . determine completely the

position and charges of all nuclei in the molecule, the operator representing kinetic energy or

the electron -electron repulsion are independent of the particular molecule.

In the second step we have to find the eigen functions and the corresponding

eigenvalues of .There is a recipe for systematically approaching the wave function of the

ground state , i-e, the state which delivers the lowest energy . This is the variational

principle, which holds a very prominent place in all quantum-chemical applications. We recall

from standard quantum mechanics that the expectation value of a particular observable

represented by the appropriate operator using any, possibly complex, wave function .

| |

The variational principle now states that the energy computed via equation (1-11) as the

expectation value of the Hamilton operator from any guessed will be an upper bound

to the true energy of the ground state i-e

| | | |

Where the equality holds if and only if is identical to . In this way we minimize the

ground state. This sure solves the problem but it doesnt simplify our work.

5

DENSITY AS A BASIC VARIABLE

Until this far, we use wave function of N particle system which is dependent on 3N spatial co-

ordinate and N spin co-ordinates, which is very difficult to solve. To simplify this problem we

use density as a basic variable which is defined by:

The operator corresponding to density is

we arrived at the conclusion that the Hamilton operator of any atomic or molecular system is

uniquely defined by N, the number of electrons, , the position of the nuclei in space, and

the charges of the nuclei. We went on by saying that once is known, we can of course only

in principle solve the Schrdinger equation.

From the properties of electron density

I. The density integrates to the number of electrons i-e

II. has maxima, that are actually even cusps, only at the positions of the nuclei

III. the density at the position of the nucleus contains information about the nuclear charge

Z

Thus, the electron density already provides all the ingredients that we identified as being

necessary for setting up the system specific Hamiltonian and it seems at least very plausible

that in fact suffices for a complete determination of all molecular properties.

6

THOMAS FERMI MODEL

Thomas and Fermi [03, 04] used the density as a basic variable and developed a theory for a

system of many particles which is known as Thomas-Fermi (TF) theory. It is the first density-

based theory for a many-electron system. This approach is only valid for infinite nuclear charge.

It only treat the kinetic energy quantum mechanically and treat and classically.

In their model Thomas and Fermi arrive at the following, very simple expression for the kinetic

energy based on the uniform electron, a fictitious model system of constant electron density.

If this is combined with classical expression of nuclear-electron attractive potential and

electron-electron repulsive potential we will get the famous Thomas Fermi expression for

energy of an atom.

The importance of this equation is not so much how well it is able to really describe the energy

of an atom, but that the energy is given completely in terms of the electron density . TF

model is a perception for how to map a density on to energy without any additional

information required. In particular, no recourse to the wave function is taken. After getting an

expression in which energy is expressed as a functional of density. To find the ground state, we

again employ variation principle. It is assumed that ground state is connected to the electron

density for which the energy according to which the Thomas Fermi equation is minimized under

the constraint.

.

For realistic systems, it yields poor quantitative predictions, even failing to reproduce some

general features of the density such as shell structure in atoms and Friedel oscillations in

solids.[05]

7

HOHNBERG AND KOHN THEOREMS:

TF model was not formulated on an exact physical basis and at that time expressing the energy

as a functional of electron density was not justify or whether a procedure of employing the

variational principle on density is allowed. It was replaced by density functional theory (DFT)

when a landmark paper by Hohnberg and Kohn [06] appeared in Physical Review. It is based on

2 fundamental theorems of Hohnberg and Kohn. These two theorems represent the foundation

of density functional theory (DFT) as an exact theory. The vital insight of HK-theory was the fact

that all the properties of a physical system can be determined from density of electron only.

The theorems are:

Theorem 1:

The electronic ground state density determines the external potential . The

density thus determines the uniquely corresponding Hamiltonian, the ground state stationary

wave function and all the electronic properties of the system

The theorem provides the proof that a physically meaning full wave functional can be

uniquely associated with a certain density is physically justified.

Kinetic energy and energy of electron-electron interaction can be expressed by use of

density. The total energy functional for a system of electrons in external potential has the

expression

[ ] [ ] [ ] [ ]

Where the external potential is

[ ]

And the potential due to electron-electron interaction is

[ ] [ ] [ ]

We can extract the classical coulomb part from [ ]

8

[ ]

Where [ ] is non-classical contribution of electron-electron interaction containing all the

effects of self-interaction correction exchange and coulomb co-relation.

Where the terms [ ] [ ] is universally valid in the sense that they are independent of

the system information i-e , while the term [ ]is dependent of the actual system.

Now the term [ ] [ ] is called HK-functional [ ].

[ ] [ ]

If we know [ ] we would have solved Schrdinger equation exactly not approximately. All

the intrinsic properties of electronics system are completely absorbed in HK-functional. It is

universal, independent of system, applied well to H-atom to gigantic molecule DNA.

Only ground state electronic density that contains the information about position and charges

of nuclei allow the mapping from density to external potential. The density of excited state

cannot be used.

Theorem 2:

The functional E[n] of total energy satisfies a variational with respect to density

The total energy E[n] reaches its minimum value E0 for correct ground state density.

[ ] [ ]

Where indicates a variation over all ground state densities of arbitrary N electrons

system. Ground state density is sufficient to obtain all properties of the system but how to find

that a certain density is really a ground state density that we are looking for.

In plane words this theorem states that [ ] , the functional that delivers the ground state

density of the system, delivers the lowest energy if and only if the input is true ground state

energy of the system. This is nothing but variational principle.

9

[ ] [ ] [ ] [ ]

Now for any trial density n(r) which satisfy the necessary boundary condition such as

and , and which is subjected to some external potential . The energy

obtained from above equation represent an upper bound to true ground state energy .

result if and only if the exact ground state density is inserted in the above equation. Now the

trial density defines its own Hamiltonian and hence its own wave function . This wave

function can now be taken as trial wave function for the Hamiltonian generated from true

external potential .Thus, we arrive at

| | [ ] [ ] [ ] [ ] | |

10

Constraints-Search Formulation

HK-theorems imply that for any arbitrary density there exists an external potential

which is called V-representable. Due to this fact it is possible to construct such densities which

are not V-representable. This implication obviously leads to formal problems which can

circumvent by more general formulation of HK-theorem called constrained-search formulation

[07].

The densities which are V-representable are called N-representable if it can be constructed

from anti-symmetric N-particles wave function. Based on so called N-representable densities, a

more general and modified scheme for [ ] minimization shows no explicit connection

to . Then the total energy functional is

[ ] [ ]| | [ ]

[ ] [ ]

The notation indicates now a minimization over all the wave functions leading

to density . Now [ ] is defined via the constraint search

[ ] [ ]| | [ ]

Now as Ritz Variational Principle comprises over all allowed wave function within its search

while constrained-search is limited to wave functions that generates . Now the

minimization runs without requiring V-representability over all wave functions. A relation of the

Ritz variational principle in with variational principle of the 2nd HK-Theorem can be established

by simply separating the search over all wave functions as it is performed in the Ritz

variation, into two separate searches that lead to a variation performed using the second HK-

Theorems.

| |

[ | | ]

11

[ | | ]

[ ]

The 1st minimum search runs over all wave functions resulting the density . The second

search after that lifts the constraint to a particular density and extends the new search

over all the densities. Though the HK-theorems and the constrained-search

formulation create a strict mathematical framework and now the existence of the total energy

functional is ensured, but one still has no practically feasible scheme to treat the N interacting

electrons at hand at this point which is solved by Kohn and Sham [08].

12

Kohn-Sham Approach

The main idea of the Kohn-sham (KS) method is to map the interacting electrons system to a

fictitious system of non-interacting electrons that produce exactly the same ground state

density and ground state energy as the interacting system. In this system, they behave as a

charge fewer particles which do not interact with each other but each of the electrons felt the

field produced by all other electrons of the system. In other words, the KS scheme represents a

mean-field theory.

The wave function [n] of the KS system is defined as the wave function that minimizes the

constrained search minimization

[ ] [ ]| | [ ] | |

Where [ ] is the kinetic energy functional of the KS system. The exchange energy by

definition is

[ ] [ ]| | [ ] [ ]

The correlation energy is

[ ] [ ]| | [ [ ]| | [ ]

Then total energy can be written as

[ ] [ ] [ ]

[ ] [ ] [ ] [ ]

Where, exchange-correlation functional is

[ ] [ ] [ ]

which contains difference of the kinetic energies plus difference of the expectation values of

the electron-electron interaction of the systems.

[ ] [ ] [ ] [ ] [ ]

13

Here the problem is in reformulating in a way that main part of the energy is treated in an exact

manner and all other unknown contributions are absorbed in the exchange-correlation

functional .This feature can be seen one of major merits of the KS-DFT.

Up till now, the original problem of the solution of Schrodingers Equation has

only transferred onto the problem of finding the exact expression for the function [ ] of

exchange-correlation, which is unknown yet.

The variational problem presented in the second HK-theorem, the ground-state energy of a

many-electron system can be obtained by minimizing the energy functional, subject to the

constraint that the number of electrons is conserved, which leads to

[ [ ] ( ]

By varying the expression (3.20) with respect to density yields the Euler-Lagrange equation of

the KS theory

[ ]

[ ]

[

[ ]

Now, defining the potential of KS in this Equation as

[

[ ]

And the Lagrange multiplier in above Equation becomes

[ ]

[ ]

With the Hartree potential

And the exchange-correlation potential

[ ]

14

To construct such energy functional there are no simple methods but it is certainly possible to

separate the exchange part from the correlation part. The same holds for the exchange and

correlation potentials as the functional derivative of the energy functional with respect to the

density

[ ]

[ ]

[ ]

[ ]

In DFT the exchange energy functional is defined by the relation [09]

[ ] [ ] [ ]| | [ ] [ ]

The term ( [ ]| | [ ] is expectation value of electron-electron interaction of KS State not

the actual electron-electron interaction energy. the difference between true electron-electron

interaction energy and expectation value ( [ ]| | [ ]) is absorbed in correlation energy,

also it contains kinetic energy differences of real and KS non-interacting systems of electrons

The correlation energy then can be

As [ ] [ ] [ ] [ ] [ ]

But [ ] [ ] [ ]

So [ ] [ ] [ ] [ ] [ ] [ ] Now put the value of [ ]

[ ]| | [ ] [ ] [ ] [ ] [ ] [ ] [ ] Cancel U[n] both sides and Rearranging we get,

[ ] [ ] [ ] [ ] [ ]| | [ ]

So by knowing the values of the four terms on the right side of this equation we can easily find

the correlation functional.

15

Approximate Exchange Co-relation Functionals

Local Density Approximation

Kohn-Sham formalism which allows an exact treatment of most of the contributions to the

electronic energy of an atomic or molecular system, including the major fraction of the kinetic

energy. All remaining unknown parts are collectively folded into the exchange-correlation

functional [ ]. These include the non-classical portion of the electron-electron interaction

along with the correction for the self-interaction and the component of the kinetic energy not

covered by the non-interacting reference system. One has to approximate the exchange co-

relation functional. One of the oldest and most known approximation (LDA), which is proposed

by Kohn and Sham themselves and it lies at the roots of DFT and KS-formalism.

The original physical system for LDA is homogeneous electron gas, means a system of electrons

which moves on a uniform background of positive charge in such a way that the entire system

becomes electrically neutral. The electrons and the volume of gas V approach to infinity, while

the electron density reaches a constant value, which remains same for each

spatial point . Inhomogeneous system of electrons such as atom, molecule or solid are treated

as homogenous system with the corresponding exchange and correlation energy

per electron at point by the associated homogenous electron gas with density at each point .

Hence, LDA approximation is most suitable for slowly varying density systems. However in solid

state physics LDA is the most employed scheme while in chemistry it has no comparable impact

in which the atoms and molecules are parameterized by a rapidly varying density. As the

homogenous electron gas is determined by its density, energy due to the exchange-correlation

within LDA approximation is calculated by integrating the parameter weighted with

the local density at each point in space

[ ]

This energy is again separated between its exchange and correlation term

[ ]

16

And LDA exchangecorrelation potential is determined by the functional derivative of above

equation

[ ]

( )

The exchange part of the energy per particle was originally derived by Dirac [10] and given

as

(

)

Where is the radius of a sphere with effective volume of an electron. However, there is no

explicit expression for the co-relation energy term. Ceperley and Alder used the quantum based

Monte Carlo scheme to calculate the total energy for a homogenous electron gas and attained

the correlation energy by subtracting the corresponding exchange and kinetic energies [11].

Various authors have presented analytical expressions of based on sophisticated

interpolation schemes. The most widely used representations of are the ones developed

by Vosko, Wilk, and Nusair (VWN) [12], 1980, while the most recent and probably also most

accurate one has been given by Perdew and Wang [13], 1992. VWN interpolated this data and

formed the LDA correlation

[

(

)

[

(

)]]

Where , , and for zero spin polarization ,

, , . Generalized Gradient Approximation In the early eighties, first successful extension to the local approximation was made. The logical first

step in that direction was the suggestion of using not only the information about density at a

particular point , but to supplement the density with information about the gradient of the charged

density in order to account for the non-homogeneity of the true electron density which is varying

fastly. So the exchange co-relation term become

17

[ ]

This approach is known as generalized gradient approximations (GGA) and generally is given by a

functional which explicitly contains density gradient . In this case the exchange and correlation

terms can also be treated separately

[ ]

[ ] [ ]

In most of the cases GGA functionals for the exchange part can be written as

[ ]

Where the argument of is a dimensionless parameter

Where F(s) is a scaling function that can have in practice a pretty complicated form. Here, stands

for the Fermi wave vector given by

.

GGA functional yields often better results than the LDA. However, some of these functionals may not

necessarily based on the new physical ideas, while some are of the semi empirical character; contains

fitting parameters and sometimes are solely constructed in order to satisfy required boundary

conditions and gain fine results in an adequate computing time. Oftenly, these are optimized in order to

gain good energy value results and do not give bearable results for the related potentials. Some of the

most important and most widespread GGAs are

Beckes exchange functional (B88) *14].

Perdew, Burke and Ernzerhofs exchange-correlation functional (PBE) [15].

Lee, Yang and Parrs correlation functional (LYP) [16].

Perdews 1986 correlation functional (P86) *17, 18].

Perdew and Wangs correlation functional (PW91) *13].

Engel and Voskos (EV) correlation functional *19].

Wu and Cohen density gradient functional (WC) [20].

18

The functional derivative of the general GGA exchange correlation functional is given by [21]

[

(

)]

(

)

[

(

)

(

)]

With two dimensionless parameters

From the above expressions it can be seen that GGA potentials contain even higher density

derivatives than the related energy functionals and together with the already some complicated

functionals of energy, the analytical equations for the potentials can become highly complicated

19

Full Potential LAPW Method The LAPW method is a very accurate scheme for the calculations of electronic structure. The

linearized augmented plane wave (LAPW) method can be implemented through WIEN2k [22]. In this

approach a special basis set is adopted to solve the problem. Basically this method is derived from the

approach of augmented-plane-wave (APW). Slater constructed APW and was motivated towards it by

the fact that close to an atomic nucleus potential and wave functions were alike to those in an atom;

nearly spherical and strongly varying. While, between the atoms both potential and wave functions

were smoother. In view of that, unit cell space is divided into two regions; non overlapping atomic

spheres and interstitial regions. The different basis expansions are applied in these regions: radial

solutions of Schrdingers equation for the atomic spheres region and plane-waves for the remaining

interstitial regions.

Where is the wave function for the system, is the cell volume and is the regular solution.

Here and are the expansion co-efficients. If is a parameter which is equal to the band

energy and is a spherical component of the potential inside the sphere, then

[

]

The particular selection of the functions was motivated by Slater who noted that plane waves are

solutions of the Schrdinger equation for constant potential, while radial functions are solutions for a

spherical potential, for equal to the eigenvalues. This approximation is known as muffin-tin

approximation and is very good for close-packed (e.g. fcc and hcp) materials. The double

representation defined by the above equation is not certain to be continuous on sphere boundaries,

as it is supposed for the kinetic energy to be well defined. Thus a constraint on the APW method is

imposed by defining in terms of the by the spherical harmonic expansion of plane waves. While

the coefficient of each lm component is matched at the sphere boundary

20

Here the origin is taken at the center of the sphere and R is sphere radius. Thus are completely

determined by the plane-wave coefficients and the energy parameters .

21

The LAPW Basis and Its Properties In LAPW method, the basis functions are the linear combinations of a radial function as

well as their energy derivatives inside the spheres, are fixed for the . The energy derivative

satisfies

[

]

In the non-relativistic case, on sphere boundaries these functions are matched to values and

derivatives of plane waves. Plane waves augmented in this way are called as the LAPW basis

functions. In the form of this basis, the wave functions

[ ]

Where are the coefficients for energy derivatives analogous to . In this scheme shape

approximations are not made and consequently such a method is often called as Full-Potential

LAPW (FP-LAPW). LAPW method is better than the APW. In the former, at a given k-point accurate

energy bands are obtained with single diagonalization, while for APW method the diagonalization is

required for each band. Moreover, LAPW basis is more flexible inside the spheres because it has two

radial functions instead of one. However, there is a price to be paid for the flexibility. This arises due

to the requirement that the basis functions must have continuous derivatives. Higher value of plane

wave cut-off is needed to achieve a given level of convergence. In the WIEN2K code, the total energy

is calculated according to the Weinert scheme [23]. The cutoff parameter controls the

convergence of the basis set and it is usually takes between 6 to 9. The is known as the muffin-tin

radii and is the smallest of all of the atomic sphere radii in the unit cell and is magnitude of the

largest K vector. Much higher angular momenta in the wave functions are required (typically 8

to 12 for accurate calculations) to satisfy the continuity conditions accurately.

22

References [01] M. Born and R. Oppenheimer, Zur Quantentheorie der Molekeln, Ann.Phys. (Leipzig) 84, 457-

484 (1927). [02+ W. Ritz, Ueber eine neue Methode zur Lsung gewisser Variations probleme der mathematischen

Physik, J. Reine Angew. Math. 135, 161 (1908). [03] L. Thomas, The calculation of atomic fields, Proc. Cambridge Phil.Soc. 23, 542 (1927) [04+ E. Fermi, Un metodo statistice per la determinazione di alcune proprieta dellatomo, Rend.

Accad. Lincei 6, 602 (1927). [05+ H. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev.136, 864 (1964). [06+ P. Hohenberg and W. Kohn, Inhomogeneous electron gas,. Phys. Rev.136, 864 (1964). [07+ M. Levy, Electron densities in search of Hamiltonians, Phys. Rev. A26, 1200 (1982). [08+ W. Kohn and L. Sham, Self-consistent equations including exchange and correlation effects,

Phys. Rev. 140, 1133 (1965). [09] R. M. Dreizler and E. K. U. Gross, Density-Functional Theory,Springer, Berlin (1990) [10+ P. A. M. Dirac, Note on exchange phenomena in the Thomas atom,Proc. Camb. Phil. Soc. 26,

376 (1930). [11+ D. M. Ceperly and B. J. Alder, Ground state of the electron gas by a stochastic method, Phys.

Rev. Lett. 45, 566 (1980). [12+ S. J. Vosko, L. Wilk, and M. Nusair, Accurate spin-dependent electron liquid correlation energies

for local spin densitycalculations: A critical analysis, Can. J. Phys. 58, 1200 (1980). [13+ J. P. Perdew, Electronic Structure of Solids, Akademie Verlag, Berlin(1991). [14+ A. D. Becke, Density-functional exchange-energy approximation with correct asymptotic

behavior, Phys. Rev. A 38, 3098 (1988). [15] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple,

Phys. Rev. Lett. 77, 3865 (1996). [16+ C. Lee, W. Yang, and R. G. Parr, Development of the Colle-Salvetti correlation-energy formula

into a functional of the electron density, Phys.Rev. B 37, 785 (1988).

23

[17+ J. P. Perdew, Density-functional approximation for the correlation energy of the inhomogeneous electron gas, Phys. Rev. B 33, 8822 (1986).

[18+ J. P. Perdew, Erratum: Density-functional approximation for the correlation energy of the

inhomogeneous electron gas, Phys. Rev. B 34, 7406(1986). [19+ E. Engle and S. H. Vosko, Exact exchange-only potentials and the virial relation as microscopic

criteria for generalized gradient approximations,Phys. Rev. B 47, 13164 (1993). [20+ Z. Wu and R. E. Cohen, More accurate generalized gradient approximation for solids, Phy. Rev.

B 73, 235116 (2006). [21+ J. P. Perdew and W. Yue, Accurate and simple density functional for the electronic exchange

energy: Generalized gradient approximation,Phys. Rev. B 33, 8800 (1986). [22+ P. Blaha, K. Schwarz, and J. Luitz, A Full Potential Linearized Augmented Planewave Package for

Calculating Crystal Properties, (TechnicalUniversity Wien, Vienna, ISBN 3-9501031-0-4, (2001). [23] M. Weinert, E. Wimmer, and A. J. Freeman, Total-energy all-electron density functional method

for bulk solids and surfaces, Phys. Rev. B 26,4571 (1982).